Exponential stability for a strongly damped wave equation with distributed internal delay

This paper investigates the exponential stability of a strongly damped wave equation subject to an internal distributed time delay. Two distinct analytical frameworks are developed: the first employs a modified energy method based on auxiliary functionals, while the second introduces a transformation that rewrites the delay term as the derivative of a convolution integral. Both approaches yield explicit decay conditions and establish exponential convergence of the energy under weaker assumptions on the damping coefficient and the delay kernel. Notably, the transformation method allows for a wider class of admissible delay weights than those permitted by classical techniques. Numerical simulations based on a finite difference scheme are presented to validate the theoretical results and to compare the performance of the two methods. These findings contribute to the qualitative theory of delayed hyperbolic systems and the stabilization of wave equations with memory effects.